Arithmetic SequenceĪn arithmetic sequence is a type of sequence where each term is obtained by adding a constant value, called the common difference, to the previous term. A sequence can be defined either explicitly, by giving a formula for each term, or recursively, by giving a rule for computing each term based on the previous terms. The index of a term refers to its position in the sequence, starting from the first term. The terms of a sequence are the values that the function outputs for each natural number input. Definition and TermsĪ sequence is a function that maps the natural numbers to a set of numbers. In this article, we will explore the different types of sequences, their properties, and their applications. They are used in many different areas of mathematics and have numerous real-world applications. Sequences are a fundamental concept in mathematics that involve ordered lists of numbers that follow a pattern or rule. Rose, A course in number theory, second edition, Clarendon Press, New York, 1994.
Parkin, "Consecutive primes in arithmetic progression," Math. Kra, "The Green-Tao theorem on arithmetic progressions in the primes: an ergodic point of view," Bull. Littlewood, "Some problems of `partitio numerorum' : III: on the expression of a number as a sum of primes," Acta Math., 44 (1923) 1-70. Guy, Unsolved problems in number theory, Springer-Verlag, 1994. Tao, "A bound for progressions of length k in the primes," (2004) Available from. GT2004a Green, Ben and Tao, Terence, "The primes contain arbitrarily long arithmetic progressions," Ann. Nelson, "Seven consecutive primes in arithmetic progression," Math. MR 1 898 760 ( Abstract available)ĭN97 H. Zimmermann, "Ten consecutive primes in arithmetic progression," Math.
van der Corput, "Über Summen von Primzahlen und Primzahlquadraten," Math. Chowla, "There exists an infinity of 3-combinations of primes in A. Ten consecutive primes in arithmetic progression by Tony Forbes.
Jens Kruse Andersen's excellent Primes in Arithmetic Progression Records.In August 2000 David Broadhurst found the smallest arithmetic progression of titanic primes of length three: (See Tony Forbes' web page for more information.) The longest known arithmetic sequence of primes is currently of length 25,Īnd continuing with common difference 366384*23#*n, found by Chermoni Raanan and Jaroslaw Wroblewski in May 2008. Obviously this is not optimal! It is conjectured įinally, in 2004, Green and Tao showed that there are indeed arbitrarily long sequences of primes and that a k-term one occurs before : Many triples of primes in arithmetic progression. In 1939, van der Corput showed that there are infinitely Such primes sequences there should be for any given length-HardyĪnd Littlewood first did this in 1922. Together these two sequences contain all of the primesĪ related question is how long of a arithmetic sequenceĬan we find all of whose members are prime.Īnswer should be arbitrarily long-but finding long sequences Then the corresponding infinite sequence contains infinitelyĪn important example of this is the following two If a 0 and d are relatively prime positive integers, In general, the terms of an arithmetic sequence withĪ n = dn+ a 0 ( n=0,1,2.). Using a common difference of 4 we get the finiteĪrithmetic sequence: 1, 5, 9, 13, 17, 21 and also the infinite sequenceġ, 5, 9, 13, 17, 21, 25, 29.
#TYPE OF SEQUENCE AND EXAMPLES PLUS#
Is a sequence (finite or infinite list) of real numbers for whichĮach term is the previous term plus a constant (called the commonĭifference). An arithmetic sequence (or arithmetic progression)
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